Slide 7:

Example 4 x = 4.30 or x = 0.70 Multiply both sides by x Take 5x over to the other side Quadratic Formula

Quadratic Theory(The Discriminant) :

Quadratic Theory(The Discriminant) Discriminant

Slide 9:

Roots:
x = 2 or x = 6 Roots:
x = 4 Roots:
x = ??

Slide 10:

From the previous examples we noted that the discriminant can help us describe the nature of the roots.
i.e. b2 - 4ac discriminates between the different types of roots.
There are three different situations as follows:

Slide 18:

Quadratic Theory(Tangents to Curves) :

Quadratic Theory(Tangents to Curves) A straight line can either:
cut a quadratic curve
touch at only one point (tangent)
or miss the curve completely

Slide 20:

To determine if a line is a tangent (i.e. cuts the curve at one point), the equation of the line is substituted into the equation of the curve. Using the resultant quadratic equation, the discriminant can be used to find the number of points of intersection.
b2- 4ac > 0 the line intersects the curve at two points
b2- 4ac = 0 the line intersects the curve at one point (the line is a tangent)
b2- 4ac < 0 the line does not intersect

Slide 27:

Quadratic Theory(Inequalities) :

Quadratic Theory(Inequalities) Opposite is the graph of y = x2 - 8x + 12
To solve x2 - 8x + 12 = 0 the roots of the equation were found (x = 2 or 6) i.e. the y co-ordinate of the points on the graph when x = 2 or x = 6 are zero.
We now consider where the y co-ordinates are not equal to zero.

Quadratic Theory(Inequalities) :

Quadratic Theory(Inequalities) To solve x2 - 8x + 12 > 0 the part of the graph above y = 0 (x axis) is considered. In this example it is when x is less than 2 but greater than 6
i.e. x < 2 or x > 6

Slide 31:

To solve x2 - 8x + 12 ≤ 0 the value of the roots and the part of the graph below y = 0 (x axis) are considered. In this example it is when x is equal to 2 or 6 and between the values of 2 and 6
i.e. 2 ≤ x ≤ 6